derivative of 1/(x(8+ln(x)))

Lösung

\frac{d}{d x} (\frac{1}{x ( 8 + ln ( x ) )})

- \frac{ln ( x ) + 9}{x ^{2} ( 8 + ln ( x ) ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{x ( 8 + ln ( x ) )})

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} ((x (8 + ln (x)))^{- 1})

Wende die Kettenregel an:

- \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} \frac{d}{d x} (x (8 + ln (x)))

= - \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} \frac{d}{d x} (x (8 + ln (x)))

\frac{d}{d x} (x (8 + ln (x))) = ln (x) + 9

= - \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} (ln (x) + 9)

Vereinfache

- \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} (ln (x) + 9) : - \frac{ln ( x ) + 9}{x ^{2} ( 8 + ln ( x ) ) ^{2}}

= - \frac{ln ( x ) + 9}{x ^{2} ( 8 + ln ( x ) ) ^{2}}

x \to - 3 lim (\frac{x + 2}{x + 3})

x \to - 3 lim (\frac{x - 2}{x + 5})

\frac{d}{d x} (x^{\frac{1}{2}} y^{\frac{1}{2}})

x \to 8 lim (x^{3} - 6 x + 5)

t an g e n t \frac{1}{x ^{2}}, at x = 3